invejfigating the Sums of Infinite Series. 
4 °' 
35 
■ + ' 
99 
v C 7 n , , i r -+&c. where n= i 2 r~A 
3*5-7*9- 1 * 5-7*9-ii-i3 1 ’ ’ ' 4* 
and the 4th differences == o ; therefore $d— 
a +3 b + l S c + 1 °5 d =3Si a +5 d +35 c +3 l 5 d=: 99 . 
-\~^ 3 c ~ s r-^ 93 d — 209., confequently a=-i, 
d'—\, therefore the fum of the given feries “ — l 
43 1 • 3-5 • 7-2.4~ 
— 1 L 1 | I ~_J_ 
3 * 5 • 7 > 2 • 3 • 4 5 • 7 • 2 • 2 . 2 7*2*4 140 
By a method fimilar to that made life of m this proportion 
may any number of fa£tors be taken from the denominators of 
thole feries delivered in part the firff, and alfo from a great 
variety of others ; but as the examples here given muff; be fuf- 
ffcient to point out the method of proceeding in all other cafes, 
v?e will proceed to the third part.. 
P A R T III. 
THE fum of every converging infinite feries, whofe 
terms ultimately become equal to nothing, may always be 
exhibited by the fum of another feries formed by colledfing 
two or more terms of the former feries into one. This 
is not true, however, where the terms of the infinite feries 
continally diverge, or converge to any affignable quantity, 
and 
